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6.2.5 Digression: $\infty $-Categories with Short Morphisms

Let $\operatorname{\mathcal{C}}$ be a category. Recall that $\operatorname{\mathcal{C}}$ is free if every morphism $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$ factors uniquely as a composition

\[ X = X_0 \xrightarrow {f_1} X_1 \xrightarrow {f_2} X_2 \rightarrow \cdots \xrightarrow {f_ n} X_ n = Y, \]

where each $f_ i$ is an indecomposable morphism of $\operatorname{\mathcal{C}}$ (see Proposition 1.2.6.11). In this case, Proposition 1.4.7.3 asserts that the inclusion map $G \hookrightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is inner anodyne, where $G$ is the $1$-dimensional simplicial set whose vertices are the objects of $\operatorname{\mathcal{C}}$ and whose nondegenerate edges are the indecomposable morphisms of $\operatorname{\mathcal{C}}$. Our goal in this section is to prove a more general result, where we relax the assumption that $\operatorname{\mathcal{C}}$ is free. First, we need a definition.

Definition 6.2.5.1. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ be a collection of morphisms of $\operatorname{\mathcal{C}}$. An $S$-optimal factorization of $f$ is a $2$-simplex

6.2
\begin{equation} \begin{gathered}\label{equation:S-optimal} \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{s} & \\ X \ar [ur]^{g} \ar [rr]^{f} & & Z } \end{gathered} \end{equation}

of $\operatorname{\mathcal{C}}$, corresponding to a morphism $\widetilde{g}: \widetilde{X} \rightarrow \widetilde{Y}$ in the $\infty $-category $\operatorname{\mathcal{C}}_{/Z}$ with the following properties:

  • The morphism $s: Y \rightarrow Z$ belongs to $S$.

  • Let $\widetilde{Y}'$ be an object of $\operatorname{\mathcal{C}}_{/Z}$ corresponding to a morphism $s': Y' \rightarrow Z$ which belongs to $S$. Then composition with $\widetilde{g}$ induces a homotopy equivalence of Kan complexes

    \[ \operatorname{Hom}_{ \operatorname{\mathcal{C}}_{/Z} }(\widetilde{Y}, \widetilde{Y}') \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}_{/Z} }( \widetilde{X}, \widetilde{Y}' ). \]

If these conditions are satisfied, we say that the diagram (6.2) is an $S$-optimal factorization of $f$.

Example 6.2.5.2. In the situation of Definition 6.2.5.1, assume that $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category. Then an $S$-optimal factorization of a morphism $f: X \rightarrow Z$ is a pair of morphisms $X \xrightarrow {g} Y \xrightarrow {s} Z$, where $s \in S$ and $s \circ g = f$, which has the following universal property: for every other pair of morphisms $X \xrightarrow {g'} Y' \xrightarrow {s'} Z$ with $s' \in S$ and $s' \circ g' = f$, there is a unique morphism $h: Y \rightarrow Y'$ satisfying $h \circ g = g'$ and $s' \circ h = s$, as indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar@ {-->}[dd]^{h} \ar [dr]^{s} & \\ X \ar [ur]^{g} \ar [dr]_{g'} & & Z \\ & Y'. \ar [ur]_{s'} & } \]

Stated more informally, the pair $(g,s)$ is universal among all factorizations of $f$ through a morphism which belongs to $S$.

Example 6.2.5.3. Let $G$ be a directed graph, let $\operatorname{\mathcal{C}}= \operatorname{Path}[G]$ denote its path category (Construction 1.2.6.1), and let $S$ be the collection of morphisms of $\operatorname{\mathcal{C}}$ which are either identity morphisms or are indecomposable. Then every morphism $f: X \rightarrow Z$ in $\operatorname{\mathcal{C}}$ admits a (unique) $S$-optimal factorization:

  • If $f = \operatorname{id}_{X}$ is an identity morphism, then its $S$-optimal factorization is given by the diagram $X \xrightarrow { \operatorname{id}_ X } X \xrightarrow { \operatorname{id}_{X} } X$.

  • If $f$ is not an identity morphism, then it admits a unique factorization $X \xrightarrow {g} Y \xrightarrow {s} Z$, where $s$ is an indecomposable morphism of $\operatorname{\mathcal{C}}$ (that is, a morphism which corresponds to an edge of the graph $G$); this factorization is $S$-optimal.

Definition 6.2.5.4. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. A class of short morphisms for $\operatorname{\mathcal{C}}$ is a collection $S$ of morphisms of $\operatorname{\mathcal{C}}$ with the following properties:

$(1)$

Every identity morphism of $\operatorname{\mathcal{C}}$ belongs to $S$.

$(2)$

For every $2$-simplex

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{f'} & \\ X \ar [ur]^{f''} \ar [rr]^{f} & & Z } \]

of the $\infty $-category $\operatorname{\mathcal{C}}$, if $f$ and $f'$ belong to $S$, then $f''$ also belongs to $S$.

$(3)$

Every morphism $f: X \rightarrow Z$ of $\operatorname{\mathcal{C}}$ admits an $S$-optimal factorization (Definition 6.2.5.1).

$(4)$

Every morphism of $\operatorname{\mathcal{C}}$ can be obtained as a composition of morphisms which belong to $S$.

Remark 6.2.5.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ be a class of short morphisms for $\operatorname{\mathcal{C}}$. Let $f: X \rightarrow Y$ and $g: X \rightarrow Y$ be morphisms of $\operatorname{\mathcal{C}}$ which are homotopic. If $f$ belongs to $S$, then $g$ also belongs to $S$. This follows by applying property $(2)$ of Definition 6.2.5.4 to a $2$-simplex

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{\operatorname{id}} & \\ X \ar [ur]^{g} \ar [rr]^{f} & & Y. } \]

Notation 6.2.5.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ be a class of short morphisms for $\operatorname{\mathcal{C}}$. We let $\operatorname{\mathcal{C}}^{\mathrm{short}} \subseteq \operatorname{\mathcal{C}}$ denote the simplicial subset consisting of those simplices $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ having the following property:

$(\ast )$

For every pair of integers $0 \leq i \leq j \leq n$, the induced morphism $\sigma (i) \rightarrow \sigma (j)$ belongs to $S$.

Note that, since $S$ contains all identity morphisms of $\operatorname{\mathcal{C}}$, condition $(\ast )$ is automatically satisfied in the case $i = j$. In particular, every vertex of $\operatorname{\mathcal{C}}$ is contained in $\operatorname{\mathcal{C}}^{\mathrm{short}}$, and an edge of $\operatorname{\mathcal{C}}$ is contained in $\operatorname{\mathcal{C}}^{\mathrm{short}}$ if and only if belongs to $S$.

Remark 6.2.5.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ be a class of short morphisms for $\operatorname{\mathcal{C}}$. Then a simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ belongs to $\operatorname{\mathcal{C}}^{\mathrm{short}}$ if and only if, for every integer $0 \leq i < n$, the morphism $\sigma (i) \rightarrow \sigma (n)$ belongs to $S$. Condition $(\ast )$ of Notation 6.2.5.6 can be deduced from this a priori weaker assumption by applying assumption $(2)$ of Definition 6.2.5.4 to the diagrams

\[ \xymatrix@R =50pt@C=50pt{ & \sigma (j) \ar [dr] & \\ \sigma (i) \ar [rr] \ar [ur] & & \sigma (n) } \]

for $i \leq j \leq n$.

Remark 6.2.5.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ be a class of short morphisms of $\operatorname{\mathcal{C}}$. Then the simplicial set $\operatorname{\mathcal{C}}^{\mathrm{short}}$ is never an $\infty $-category, except in the trivial situation where $S$ is the class of all morphisms of $\operatorname{\mathcal{C}}$ (in which case we have $\operatorname{\mathcal{C}}^{\mathrm{short}} = \operatorname{\mathcal{C}}$). However, for every object $Z \in \operatorname{\mathcal{C}}$, the simplicial set $\operatorname{\mathcal{C}}^{\mathrm{short}}_{/Z} = ( \operatorname{\mathcal{C}}^{\mathrm{short}})_{/Z}$ is an $\infty $-category, since it can be identified with the full subcategory of $\operatorname{\mathcal{C}}_{/Z}$ spanned by those morphisms $s: Y \rightarrow Z$ which belong to $S$.

Remark 6.2.5.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $S$ be a class of short morphisms of $\operatorname{\mathcal{C}}$, and let $f: X \rightarrow Z$ be a morphism of $\operatorname{\mathcal{C}}$, which we identify with an object $\widetilde{X}$ of the slice $\infty $-category $\operatorname{\mathcal{C}}_{/Z}$. Then an $S$-optimal factorization of $f$ can be viewed as a morphism $\widetilde{X} \rightarrow \widetilde{Y}$ in $\operatorname{\mathcal{C}}_{/Z}$ which exhibits $\widetilde{Y}$ as a $\operatorname{\mathcal{C}}^{\mathrm{short}}_{/Z}$-reflection of $\widetilde{X}$, in the sense of Definition 6.2.2.1. Consequently, condition $(3)$ of Definition 6.2.5.4 is equivalent to the requirement that the full subcategory $\operatorname{\mathcal{C}}^{\mathrm{short}}_{/Z} \subseteq \operatorname{\mathcal{C}}_{/Z}$ is reflective, for each object $Z \in \operatorname{\mathcal{C}}$.

We can now state our main result.

Theorem 6.2.5.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ be a class of short morphisms for $\operatorname{\mathcal{C}}$. Then the inclusion map $\operatorname{\mathcal{C}}^{\mathrm{short}} \hookrightarrow \operatorname{\mathcal{C}}$ is an inner anodyne morphism of simplicial sets.

Example 6.2.5.11. Let $G$ be a directed graph and let $\operatorname{\mathcal{C}}= \operatorname{Path}[G]$ denote its path category (Construction 1.2.6.1). Let $S$ be the collection of morphisms of $\operatorname{\mathcal{C}}$ which are either identity morphisms or are indecomposable. Then $S$ is a class of short morphisms for the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ (the existence of $S$-optimal factorizations follows from Example 6.2.5.3, and the remaining requirements are immediate from the definitions). Moreover, the simplicial set $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})^{\mathrm{short}}$ can be identified with the directed graph $G$ (regarded as a $1$-dimensional simplicial set; see ยง1.1.5). Applying Theorem 6.2.5.10 in this case, we recover the statement that the inclusion map $G \hookrightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is inner anodyne (Proposition 1.4.7.3).

Our proof of Theorem 6.2.5.10 will require some auxiliary constructions.

Notation 6.2.5.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $S$ be a class of short morphisms for $\operatorname{\mathcal{C}}$, and let $f: X \rightarrow Y$ be a morphism of $\operatorname{\mathcal{C}}$. We let $\ell (f)$ denote the smallest integer $n$ for which there exists an $n$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ for which the composition

\[ \Delta ^1 \rightarrow \operatorname{N}_{\bullet }( \{ 0,n \} ) \hookrightarrow \Delta ^ n \xrightarrow {\sigma } \operatorname{\mathcal{C}}. \]

coincides with $f$. Note that condition $(4)$ of Definition 6.2.5.4 guarantees that $\ell (f) < \infty $. We will refer to $\ell (f)$ as the $S$-length of $f$. Note that $\ell (f) = 0$ if and only if $f$ is an identity morphism of $\operatorname{\mathcal{C}}$, and $\ell (f) \leq 1$ if and only if $f$ belongs to $S$.

Lemma 6.2.5.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $S$ be a class of short morphisms of $\operatorname{\mathcal{C}}$, and suppose we are given a $2$-simplex

6.3
\begin{equation} \begin{gathered}\label{equation:S-length-decrease} \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{s} & \\ X \ar [ur]^{g} \ar [rr]^{f} & & Z, } \end{gathered} \end{equation}

of $\operatorname{\mathcal{C}}$, where $s$ belongs to $S$. Then:

$(a)$

If $\ell (f) \geq 1$, then $\ell (g) \leq \ell (f)$.

$(b)$

If $\ell (f) \geq 2$ and the factorization (6.3) is $S$-optimal, then $\ell (g) = \ell (f) - 1$.

Proof. We prove $(a)$ and $(b)$ by simultaneous induction on the length $n = \ell (f)$. If $n = 1$, then assertion $(a)$ follows from condition $(2)$ of Definition 6.2.5.4 and assertion $(b)$ is vacuous. We therefore assume that $n \geq 2$. We first prove $(b)$. Choose a factorization

\[ \xymatrix@R =50pt@C=50pt{ & Y' \ar [dr]^{s'} & \\ X \ar [ur]^{g'} \ar [rr]^{f} & & Z, } \]

where $s' \in S$ and $\ell (g') \leq n-1$. If the factorization (6.3) is $S$-optimal, then we can choose a $3$-simplex

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dd]^{h} \ar [dr]^{s} & \\ X \ar [ur]^{g} \ar [dr]^{g'} & & Z \\ & Y'. \ar [ur]^{s'} & } \]

Condition $(2)$ of Definition 6.2.5.4 guarantees that $h$ belongs to $S$. Our inductive hypothesis guarantees that the left half of the diagram satisfies assertion $(a)$; so that $\ell (g) \leq \ell (g') = \ell (f) - 1$. The reverse inequality follows immediately from the definition.

We now prove $(a)$. Choose an $S$-optimal factorization

\[ \xymatrix@R =50pt@C=50pt{ & Y'' \ar [dr]^{s''} & \\ X \ar [ur]^{g''} \ar [rr]^{f} & & Z. } \]

It follows from the preceding argument that $\ell (g'') = n-1$. We then have a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ & Y'' \ar [dd]^{j} \ar [dr]^{s''} & \\ X \ar [ur]^{g''} \ar [dr]^{g} & & Z \\ & Y, \ar [ur]^{s} & } \]

and condition $(2)$ of Definition 6.2.5.4 guarantees that $j$ belongs to $S$. We therefore obtain $\ell (g) \leq \ell (g'') + \ell (j) \leq (n-1) + 1 = n$, as desired. $\square$

Notation 6.2.5.14. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ be a class of short morphisms of $\operatorname{\mathcal{C}}$. For every $n$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$, we let $\mathrm{pr}(\sigma )$ denote the smallest nonnegative integer $p$ such that, for $p \leq q \leq n$, the morphism $\sigma (q) \rightarrow \sigma (n)$ belongs to $S$. We will refer to $\mathrm{pr}( \sigma )$ as the priority of $\sigma $. This definition has the following properties:

  • The simplex $\sigma $ has priority $0$ if and only if it belongs to the simplicial subset $\operatorname{\mathcal{C}}^{\mathrm{short}}$ of Notation 6.2.5.6 (see Remark 6.2.5.7).

  • For each $0 \leq i \leq n$, the face $\tau = d_ i(\sigma )$ satisfies $\mathrm{pr}( \tau ) \leq \mathrm{pr}(\sigma )$. The inequality is strict if $i < \mathrm{pr}(\sigma )$, and equality holds if $\mathrm{pr}(\sigma ) \leq i < n$.

  • For each $0 \leq i \leq n$, the degenerate simplex $\tau = s_ i(\sigma )$ satisfies

    \[ \mathrm{pr}( \tau ) = \begin{cases} \mathrm{pr}( \sigma ) + 1 & \text{ if $0 \leq i < \mathrm{pr}( \sigma )$ } \\ \mathrm{pr}(\sigma ) & \text{ if $\mathrm{pr}(\sigma ) \leq i \leq n$} \end{cases} \]

In the situation of Notation 6.2.5.14, suppose that $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category. An $n$-simplex of $\operatorname{\mathcal{C}}$ can then be viewed as a diagram

\[ X_0 \rightarrow X_1 \rightarrow \cdots \rightarrow X_{n}, \]

whose transition maps we denote by $f_{j,i}: X_ i \rightarrow X_ j$. If $\sigma $ does not belong to $\operatorname{\mathcal{C}}^{\mathrm{short}}$, then the priority of $\sigma $ is the smallest integer $p$ for which the morphism $f_{n, p-1}: X_{p-1} \rightarrow X_ n$ does not belong to $S$. Then $f_{n,p-1}$ admits an $S$-optimal factorization $X_{p-1} \xrightarrow {g} Y \xrightarrow {s} X_{n}$. Since the morphism $f_{n,p}: X_ p \rightarrow X_ n$ belongs to $S$, there is a unique morphism $h: Y \rightarrow X_ p$ for which the diagram

\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dd]^{h} \ar [dr]^{s} & \\ X_{p-1} \ar [ur]^{g} \ar [dr]_{f_{p,p-1}} & & X_{n} \\ & X_ p \ar [ur]_{f_{n,p}} & } \]

is commutative. The diagram

\[ X_0 \rightarrow \cdots \rightarrow X_{p-1} \xrightarrow {g} Y \xrightarrow { h } X_{p} \rightarrow \cdots \rightarrow X_{n} \]

then determines an $(n+1)$-simplex $\sigma ^{+}$ of $\operatorname{\mathcal{C}}$ having priority $p$, which satisfies $d_ p( \sigma ^{+} ) = \sigma $. To prove Theorem 6.2.5.10, we will extend the construction $\sigma \mapsto \sigma ^{+}$ to the case where $\operatorname{\mathcal{C}}$ is an $\infty $-category.

Lemma 6.2.5.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ be a class of short morphisms of $\operatorname{\mathcal{C}}$. Then there is a function which associates to each $n$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ which does not belong to $\operatorname{\mathcal{C}}^{\mathrm{short}}$ an $(n+1)$-simplex $\sigma ^{+}$ of $\operatorname{\mathcal{C}}$, which has the following properties:

$(1)$

The face operators satisfy

\[ d_{i}( \sigma ^{+} ) = \begin{cases} \sigma & \text{ if $i = \mathrm{pr}( \sigma )$ } \\ d_{i-1}(\sigma )^{+} & \text{ if $\mathrm{pr}(\sigma ) < i \leq n$.} \end{cases} \]
$(2)$

Let $\sigma = s_ j(\tau )$ be a degenerate $n$-simplex of $\operatorname{\mathcal{C}}$. Then

\[ \sigma ^{+} = \begin{cases} s_ j( \tau ^{+} ) & \text{ if $0 \leq j < \mathrm{pr}(\tau )$ } \\ s_{j+1}( \tau ^{+} ) & \text{ if $\mathrm{pr}(\tau ) \leq j < n$. } \end{cases} \]
$(3)$

If $\sigma = \tau ^{+}$ for some $(n-1)$-simplex $\tau $ of $\operatorname{\mathcal{C}}$ having priority $p > 0$, then $\sigma ^{+} = s_ p( \sigma )$.

$(4)$

If $\mathrm{pr}(\sigma ) = n$, then the $2$-simplex

\[ \Delta ^2 \simeq \operatorname{N}_{\bullet }( \{ n-1 < n < n+1 \} ) \hookrightarrow \Delta ^{n+1} \xrightarrow { \sigma ^{+} } \operatorname{\mathcal{C}} \]

is an $S$-optimal factorization.

Exercise 6.2.5.16. Prove Lemma 6.2.5.15 in the special case where $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category.

We defer the (somewhat tedious) proof of Lemma 6.2.5.15 until the end of this section.

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

\[ \operatorname{\mathcal{C}}^{\mathrm{short}} = \operatorname{\mathcal{C}}^{\leq 0} \hookrightarrow \operatorname{\mathcal{C}}^{\leq 1} \hookrightarrow \operatorname{\mathcal{C}}^{\leq 2} \hookrightarrow \cdots \]

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

\[ \operatorname{\mathcal{C}}^{\leq p-1} = \operatorname{\mathcal{C}}^{\leq p}(0) \hookrightarrow \operatorname{\mathcal{C}}^{\leq p}(1) \hookrightarrow \operatorname{\mathcal{C}}^{\leq p}(2) \hookrightarrow \operatorname{\mathcal{C}}^{\leq p}(3) \hookrightarrow \cdots \]

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 $\sigma ^{+} = s_ i( \sigma _0^{+} )$ 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 ocntained 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 5.4.1.35, we can choose a well-ordering on $T$ for which the complexity function

\[ c: T \rightarrow \operatorname{\mathbf{Z}}_{\geq 0} \quad \quad t \mapsto c(t) \]

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

6.10
\begin{equation} \begin{gathered}\label{equation:short-morphisms} \xymatrix { L_{t} \ar [r] \ar [d] & \operatorname{\mathcal{C}}^{\leq p}_{< t}(n) \ar [d] \\ \Delta ^{n+1} \ar [r]^{ \sigma _{t}^{+} } & \operatorname{\mathcal{C}}^{\leq p}_{\leq t}(n). } \end{gathered} \end{equation}

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 3.1.2.10). This is equivalent to the following more concrete assertion:

$(\ast _{t})$

For $0 \leq i \leq n+1$, the $n$-simplex $d_{i}( \sigma _{t}^{+} )$ 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 $d_ i( \sigma _{t}^{+} )$ 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 $d_ i( \sigma _{t}^{+} )$ 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'

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

  • Suppose that $i = n+1$ and set $\tau = d_{i}( \sigma _{t}^{+} )$; we wish to show that $\tau $ is contained in $\operatorname{\mathcal{C}}^{\leq p}_{< t}(n-1)$. 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 $\tau = \sigma _{t'} = d_ p( \sigma _{t'}^{+} )$ is contained in $\operatorname{\mathcal{C}}^{\leq p}_{\leq t'}(n) \subseteq \operatorname{\mathcal{C}}^{\leq p}_{< t}(n)$.

$\square$

Proof of Lemma 6.2.5.15. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ be a class of short morphisms for $\operatorname{\mathcal{C}}$. Our construction proceeds by recursion. Fix an integer $n \geq 0$. Assume that we have constructed a function $\tau \mapsto \tau ^{+}$ on simplices of $\operatorname{\mathcal{C}}$ having dimension $< n$ and priority $> 0$, satisfying conditions $(1)$ through $(4)$ of Lemma 6.2.5.15. Let $\sigma $ be an $n$-simplex of $\operatorname{\mathcal{C}}$ having priority $> 0$; we wish to show that there is an $(n+1)$-simplex $\sigma ^{+}$ which also satisfies conditions $(1)$ through $(4)$. Let us say that $\sigma $ of $\operatorname{\mathcal{C}}$ is free if it is not of the form $\tau ^{+}$, where $\tau $ is an $(n-1)$-simplex of priority $> 0$. We divide the construction into three cases:

$(a)$

The $n$-simplex $\sigma $ is not free.

$(b)$

The $n$-simplex $\sigma $ is free and degenerate.

$(c)$

The $n$-simplex $\sigma $ is free and nondegenerate.

We begin with case $(a)$. Assume that $\sigma = \tau ^{+}$, where $\tau $ is an $(n-1)$-simplex of $\operatorname{\mathcal{C}}$ having priority $p > 0$. It follows from our inductive hypothesis that $\sigma $ has the same priority $p$, and that $\tau = d_ p(\sigma )$. In particular, $\tau $ is uniquely determined by $\sigma $. In this case, we define $\sigma ^{+} = s_ p(\sigma )$, so that condition $(3)$ is satisfied by construction. Since $p \leq n-1 < n$, condition $(4)$ is vacuous. Note that the faces $d_ p( \sigma ^{+} )$ and $d_{p+1}( \sigma ^{+} )$ coincide with $\sigma = \tau ^{+} = d_ p( \sigma )^{+}$, so that condition $(1)$ is satisfied for $i = p$ and $i = p+1$. For $p+1 < i \leq n$, we compute

\begin{eqnarray*} d_ i( \sigma ^{+} ) & = & d_ i( s_ p( \tau ^{+} ) ) \\ & = & s_ p( d_{i-1}( \tau ^{+} ) ) \\ & = & s_ p( d_{i-2}(\tau )^{+} ) \\ & = & (d_{i-2}(\tau )^{+})^{+} \\ & = & d_{i-1}( \tau ^{+} )^{+} \\ & = & d_{i-1}(\sigma )^{+}. \end{eqnarray*}

It remains to verify condition $(2)$. Suppose that $\sigma = s_ j(\sigma ')$ for some $(n-1)$-simplex $\sigma '$ of $\operatorname{\mathcal{C}}$. Note that, since $\sigma $ has priority $p$, we must have $j \neq p-1$ (see Notation 6.2.5.14). We first consider the case $j < p-1$, so that $\sigma '$ has priority $p-1$. In this case, we wish to show that $\sigma ^{+} = s_ j( \sigma '^{+})$. Set $\tau ' = d_{p-1}( \sigma ' )$. We then have

\[ \tau = d_ p( \sigma ) = d_ p( s_ j( \sigma ' ) ) = s_{j}( d_{p-1}( \sigma ' ) ) = s_ j(\tau ' ), \]

so that $\sigma = \tau ^{+} = s_ j( \tau '^{+} )$. Applying the face map $d_ j$, we obtain $\sigma ' = \tau '^{+}$, so that $\sigma '^{+} = s_{p-1}( \sigma ' )$. The desired result now follows from the calculation

\[ \sigma ^{+} = s_ p( \sigma ) = s_ p( s_ j( \sigma ' ) ) = s_ j( s_{p-1}( \sigma ' ) ) = s_ j( \sigma '^{+} ). \]

We now treat the case $j \geq p$, so that $\sigma '$ has priority $p$. In this case, we wish to show that $\sigma ^{+} = s_{j+1}( \sigma '^{+} )$. If $j = p$, this follows from the calculation

\begin{eqnarray*} \sigma ^{+} & = & s_ p( \sigma ) \\ & = & s_ p( s_ p( \sigma ' )) \\ & = & s_{p+1}( s_ p( \sigma ' ) ) \\ & = & s_{p+1}(\sigma ) \\ & = & s_{p+1}( \tau ^{+} ). \end{eqnarray*}

Let us therefore assume that $j > p$, and set $\tau ' = d_{p}( \sigma ' )$. We then have

\[ \tau = d_ p( \sigma ) = d_ p( s_ j( \sigma ' ) ) = s_{j-1}( d_{p}( \sigma ' ) ) = s_{j-1}(\tau ' ), \]

so that $\sigma = \tau ^{+} = s_ j( \tau '^{+} )$. Applying the face map $d_ j$, we obtain $\sigma ' = \tau '^{+}$, so that $\sigma '^{+} = s_{p}( \sigma ' )$. The desired result now follows from the calculation

\[ \sigma ^{+} = s_ p( \sigma ) = s_ p( s_ j( \sigma ' ) ) = s_{j+1}( s_{p}( \sigma ' ) ) = s_{j+1}( \sigma '^{+} ). \]

This completes our treatment of case $(a)$.

We now consider case $(b)$. Assume that $\sigma $ is a free simplex of $\operatorname{\mathcal{C}}$ of the form $s_ j( \tau )$. Choose $j$ as small as possible and let $p$ be the priority of $\tau $. We first treat the case where $j < p$, so that $\sigma $ has priority $p+1$ (see Notation 6.2.5.14). In this case, we define $\sigma ^{+} = s_ j( \tau ^{+} )$, so that

\[ d_{p+1}( \sigma ^{+} ) = d_{p+1}( s_ j( \tau ^{+} ) ) = s_ j( d_ p( \tau ^{+} ) ) = s_ j( \tau ) = \sigma . \]

For $p+1 < i \leq n$, a similar calculation gives

\begin{eqnarray*} d_ i( \sigma ^{+} ) & = & d_ i( s_ j( \tau ^{+} ) ) \\ & = & s_ j( d_{i-1}( \tau ^{+} ) ) \\ & = & s_ j( d_{i-2}(\tau )^{+} ) \\ & = & s_ j(d_{i-2}(\tau ))^{+} \\ & = & d_{i-1}( s_ j(\tau ) )^{+} \\ & = & d_{i-1}( \sigma )^{+}, \end{eqnarray*}

which proves $(1)$.

To verify $(2)$, suppose that $\sigma = s_{j'}( \tau ' )$, for some $(n-1)$-simplex $\tau '$ of $\operatorname{\mathcal{C}}$. Note that we must have $j' \geq j$. Since $\sigma $ has priority $p+1$, we also have $j' \neq p$. Assume first that $j' < p$, so that $\tau '$ has priority $p$. In this case, we wish to show that $\sigma ^{+} = s_{j'} ( \tau '^{+} )$. If $j' = j$, this is immediate. We may therefore assume that $j' > j$, so that we can write $\tau ' = s_ j( \tau '' )$ and $\tau = s_{j'-1}( \tau '' )$ for some unique $(n-2)$-simplex $\tau ''$ of $\operatorname{\mathcal{C}}$. In this case, the desired result follows from the calculation

\begin{eqnarray*} \sigma ^{+} & = & s_ j( \tau ^{+}) \\ & = & s_ j( s_{j'-1}(\tau '')^{+} ) \\ & = & s_ j( s_{j'-1}( \tau ''^+ )) \\ & = & s_{j'} s_ j( \tau ''^{+} ) \\ & = & s_{j'} (s_ j(\tau '')^{+}) \\ & = & s_{j'} (\tau '^{+} ). \end{eqnarray*}

If $j' > p$, then $\tau '$ instead has priority $p+1$, and the desired result follows instead from the calculation

\begin{eqnarray*} \sigma ^{+} & = & s_ j( \tau ^{+}) \\ & = & s_ j( s_{j'-1}(\tau '')^{+} ) \\ & = & s_ j( s_{j'}( \tau ''^+ )) \\ & = & s_{j'+1} s_ j( \tau ''^{+} ) \\ & = & s_{j'+1} (s_ j(\tau '')^{+}) \\ & = & s_{j'+1} (\tau '^{+} ). \end{eqnarray*}

Condition $(3)$ is vacuous (since we have assumed that $\sigma $ is free). To prove $(4)$, we note that if $\sigma $ has priority $n$, then $\tau $ has priority $(n-1)$; the desired result now follows from the observation that the restriction of $\sigma ^{+}$ to $\operatorname{N}_{\bullet }( \{ n-1 < n < n+1 \} )$ coincides with the restriction of $\tau ^{+}$ to $\operatorname{N}_{\bullet }( \{ n-2 < n-1 < n \} )$, and is therefore an $S$-optimal factorization. This completes the construction in the case $j < p$.

We now treat the case $j \geq p$, so that the simplex $\sigma = s_ j(\tau )$ has priority $p$. In this case, we set $\sigma ^{+} = s_{j+1}(\tau ^{+} )$. Condition $(3)$ again vacuous (since $\sigma $ is assumed to be free), and condition $(4)$ is vacuous since $p < n$. We next prove $(1)$. Note that we have

\[ d_ p( \sigma ^{+} ) = d_ p( s_{j+1}( \tau ^{+} ) ) = s_{j}( d_ p( \tau ^{+} ) ) = s_ j( \tau ) = \sigma . \]

To complete the proof of $(1)$, we must show that $d_ i( \sigma ^{+} ) = d_{i-1}(\sigma )^{+}$ for $p < i \leq n$. For $i \leq j$, this follows from the calculation

\begin{eqnarray*} d_ i( \sigma ^{+} ) & = & d_ i( s_{j+1}( \tau ^{+} ) ) \\ & = & s_ j( d_ i( \tau ^{+} ) ) \\ & = & s_ j( d_{i-1}(\tau )^{+} ) \\ & = & s_{j-1}( d_{i-1}(\tau ) )^{+} \\ & = & d_{i-1}( s_ j( \tau ) )^{+} \\ & = & d_{i-1}(\sigma )^{+}. \end{eqnarray*}

For $j+2 < i \leq n$, it follows instead from the calculation

\begin{eqnarray*} d_ i( \sigma ^{+} ) & = & d_ i( s_{j+1}( \tau ^{+} ) ) \\ & = & s_{j+1}( d_{i-1}( \tau ^{+} ) ) \\ & = & s_{j+1}( d_{i-2}(\tau )^{+} ) \\ & = & s_{j}( d_{i-2}(\tau ) )^{+} \\ & = & d_{i-1}( s_ j( \tau ) )^{+} \\ & = & d_{i-1}(\sigma )^{+}. \end{eqnarray*}

It will therefore suffice to treat the case $i \in \{ j+1, j+2 \} $, in which case we have

\begin{eqnarray*} d_ i( \sigma ^+ ) & = & d_ i( s_{j+1}( \tau ^{+} ) ) \\ & = & \tau ^{+} \\ & = & d_{i-1}( s_ j( \tau ) )^{+} \\ & = & d_{i-1}( \sigma )^{+}. \end{eqnarray*}

To verify condition $(2)$, suppose that $\sigma = s_{j'}( \tau ' )$. By construction, we then have $j' \geq j \geq p$, so that the simplex $\tau '$ has priority $p$. We wish to show that $\sigma ^{+} = s_{j'+1}( \tau '^{+} )$. If $j' = j$, this is immediate. We may therefore assume that $j' > j$, so that we can write $\tau ' = s_ j( \tau '' )$ and $\tau = s_{j'-1}( \tau '' )$ as above. In this case, the desired result follows from the calculation

\begin{eqnarray*} \sigma ^{+} & = & s_{j+1}( \tau ^{+}) \\ & = & s_{j+1}( s_{j'-1}(\tau '')^{+} ) \\ & = & s_{j+1}( s_{j'}( \tau ''^+ )) \\ & = & s_{j'+1} s_{j+1}( \tau ''^{+} ) \\ & = & s_{j'+1} (s_{j}(\tau '')^{+}) \\ & = & s_{j'+1} (\tau '^{+} ). \end{eqnarray*}

This completes the treatment of case $(b)$.

We now consider case $(c)$. For the remainder of the proof, we assume that the simplex $\sigma $ is free and nondegenerate, of priority $p > 0$. Let us decompose $\Delta ^{n+1}$ as a join $\Delta ^{p-1} \star \Delta ^{n-p} \star \{ z\} $. In what follows, we write $x$ for the final vertex of $\Delta ^{p-1}$ (corresponding to the element $p-1 \in [n+1]$) and $y$ for the initial vertex of $\Delta ^{n-p}$ (corresponding to the element $p \in [n+1]$). Note that the $n$-simplices $\sigma $ and $\{ d_ i(\sigma )^{+} \} _{ p \leq i < n}$ determine a morphism of simplicial sets $\sigma ^{\dagger }: \Delta ^{p-1} \star \operatorname{\partial \Delta }^{n-p} \star \{ z\} \rightarrow \operatorname{\mathcal{C}}$. Unwinding the definitions, we see that an $(n+1)$-simplex $\sigma ^{+}$ of $\operatorname{\mathcal{C}}$ satisfying condition $(1)$ can be identified with an extension of $\sigma ^{\dagger }$ to the join $\Delta ^{p-1} \star \Delta ^{n-p} \star \{ z\} \simeq \Delta ^{n+1}$. We wish to show that such an extension can always be found, which additional satisfies condition $(4)$ in the case $p=n$ (note that conditions $(2)$ and $(3)$ are vacuous, by virtue of our assumption that $\sigma $ is free and nondegenerate).

Let $\overline{\sigma }^{\dagger }$ denote the restriction of $\sigma ^{\dagger }$ to $\{ x\} \star \operatorname{\partial \Delta }^{n-p} \star \{ z\} $. Since the inclusion $\{ x\} \hookrightarrow \Delta ^{p-1}$ is right anodyne (Example 4.3.7.11), it will suffice to show that $\overline{\sigma }^{\dagger }$ can be extended to an $(n+2-p)$-simplex $\overline{\sigma }^{+}$ of $\operatorname{\mathcal{C}}$, having the additional property that $\overline{\sigma }^{+}$ is an $S$-optimal factorization in the case $p=n$. If $p = n$, the existence of $\overline{\sigma }^{+}$ follows from our assumption that $S$ is a class of short morphisms for $\operatorname{\mathcal{C}}$. We therefore assume that $p < n$. Set $Z = \sigma ^{\dagger }(z)$, so that we can identify $\overline{\sigma }^{\dagger }$ with a morphism of simplicial sets $\rho _0: \Lambda ^{n+1-p}_{0} \rightarrow \operatorname{\mathcal{C}}_{/Z}$; we wish to extend $\rho _0$ to an $(n+1-p)$-simplex of $\operatorname{\mathcal{C}}_{/Z}$. For $0 < i \leq n+1-p$, the image $\rho _0(i)$ belongs to the full subcategory $\operatorname{\mathcal{C}}_{/Z}^{\mathrm{short}} \subseteq \operatorname{\mathcal{C}}_{/Z}$. By virtue of Proposition 6.2.2.7, it will suffice to show that the restriction of $\rho _0$ to $\Delta ^1$ exhibits $\rho _0(1)$ as a $\operatorname{\mathcal{C}}_{/Z}^{\mathrm{short}}$-reflection of $\rho _0(0)$. This is equivalent to the assertion that the $2$-simplex

\[ \xymatrix { & \sigma ^{\dagger }(y) \ar [dr] & \\ \sigma ^{\dagger }(x) \ar [rr] \ar [ur] & & \sigma ^{\dagger }(z) } \]

is an $S$-optimal factorization of the lower horizontal morphism (Remark 6.2.5.9), which follows from our inductive hypothesis. $\square$