# Kerodon

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### 1.5.7 Universality of Path Categories

Let $G$ be a directed graph, let $G_{\bullet }$ denote the associated $1$-dimensional simplicial set (see Proposition 1.1.6.9), and let $\operatorname{Path}[G]$ denote the path category of $G$ (Construction 1.3.7.1). There is an evident map of simplicial sets $u: G_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] )$. By virtue of Proposition 1.3.7.5, this map exhibits $\operatorname{Path}[G]$ as the homotopy category of the simplicial set $G_{\bullet }$. In other words, the path category $\operatorname{Path}[G]$ is universal among categories $\operatorname{\mathcal{C}}$ which are equipped with a $G_{\bullet }$-indexed diagram (see Definition 1.5.2.1). Our goal in this section is to establish a variant of this statement in the setting of $\infty$-categories:

Theorem 1.5.7.1. Let $G$ be a directed graph and let $\operatorname{\mathcal{C}}$ be an $\infty$-category. Then composition with the map of simplicial sets $u: G_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ induces a trivial Kan fibration of simplicial sets $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Path}[G] ), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( G_{\bullet }, \operatorname{\mathcal{C}})$.

More informally, Theorem 1.5.7.1 asserts that any $G$-indexed diagram in an $\infty$-category $\operatorname{\mathcal{C}}$ admits an essentially unique extension to a functor of $\infty$-categories $\operatorname{N}_{\bullet }( \operatorname{Path}[G] ) \rightarrow \operatorname{\mathcal{C}}$.

Example 1.5.7.2. Let $G$ be the directed graph depicted in the diagram

$\xymatrix@R =50pt@C=50pt{ \bullet \ar [r] & \bullet \ar [r] & \bullet . }$

Then the map $u: G_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ can be identified with the inclusion of simplicial sets $\Lambda ^{2}_{1} \hookrightarrow \Delta ^2$. In this case, Theorem 1.5.7.1 reduces to the statement that the map

$\operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Lambda ^{2}_{1}, \operatorname{\mathcal{C}})$

is a trivial Kan fibration, which is equivalent to the assumption that $\operatorname{\mathcal{C}}$ is an $\infty$-category by virtue of Theorem 1.5.6.1.

We will deduce Theorem 1.5.7.1 from the following more precise assertion.

Proposition 1.5.7.3. Let $G$ be a directed graph. Then the map of simplicial sets $u: G_{\bullet } \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ is inner anodyne (Definition 1.5.6.4).

Remark 1.5.7.4. Let $G$ be a directed graph and let $\operatorname{\mathcal{C}}$ be an ordinary category. Combining Proposition 1.5.7.3 with Variant 1.5.6.8, we deduce that the canonical map

$\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }( \operatorname{Path}[G] ), \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( G_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )$

is bijective. Combining this observation with Proposition 1.3.3.1, we obtain a bijection

$\operatorname{Hom}_{\operatorname{Cat}}( \operatorname{Path}[G], \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( G_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ).$

Allowing $\operatorname{\mathcal{C}}$ to vary, we recover the assertion that $u: G_{\bullet } \rightarrow \operatorname{N}_{\bullet }(\operatorname{Path}[G] )$ exhibits $\operatorname{Path}[G]$ as the homotopy category of $G_{\bullet }$ (Proposition 1.3.7.5).

Let us first show that Proposition 1.5.7.3 implies Theorem 1.5.7.1.

Lemma 1.5.7.5. Let $f: X \hookrightarrow Y$ and $f': X' \hookrightarrow Y'$ be monomorphisms of simplicial sets. If $f$ is inner anodyne, then the induced map

$u_{f,f'}: (Y \times X' ) \coprod _{ (X \times X')} (X \times Y' ) \hookrightarrow Y \times Y'$

is inner anodyne.

Proof. Let us regard the morphism $f': X' \hookrightarrow Y'$ as fixed. Let $T$ be the collection of all morphisms $f: X \rightarrow Y$ for which the map $u_{f,f'}$ is inner anodyne. Then $T$ is weakly saturated. To prove Lemma 1.5.7.5, we must show that $T$ contains all inner anodyne morphisms of simplicial sets. By virtue of Lemma 1.5.6.9, it will suffice to show that $T$ contains every morphism of the form

$u_{i,j}: (B \times \Lambda ^2_1) \coprod _{A \times \Lambda ^2_1 } (A \times \Delta ^2) \subseteq B \times \Delta ^2,$

where $i: A \hookrightarrow B$ is a monomorphism of simplicial sets and $j: \Lambda ^2_1 \hookrightarrow \Delta ^2$ is the inclusion. Setting

$A' = (B \times X') \coprod _{ (A \times X' )} ( A \times Y') \quad \quad B' = B \times Y',$

we are reduced to the problem of showing that the map

$u_{i',j}: (B' \times \Lambda ^2_1) \coprod _{A' \times \Lambda ^2_1 } (A' \times \Delta ^2) \subseteq B' \times \Delta ^2,$

is inner anodyne, which follows from Lemma 1.5.6.9. $\square$

Proposition 1.5.7.6. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $f: X \hookrightarrow Y$ be an inner anodyne morphism of simplicial sets. Then the induced map $p: \operatorname{Fun}( Y, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( X, \operatorname{\mathcal{C}})$ is a trivial Kan fibration.

Proof. To show that $p$ is a trivial Kan fibration, it will suffice to show that it is weakly right orthogonal to every monomorphism of simplicial sets $f': X' \hookrightarrow Y'$. This is equivalent to the assertion that every map of simplicial sets

$g_0: (Y \times X' ) \coprod _{ (X \times X')} (X \times Y' ) \rightarrow \operatorname{\mathcal{C}}$

can be extended to a map $g: Y \times Y' \rightarrow \operatorname{\mathcal{C}}$. This follows from Proposition 1.5.6.7, since $\operatorname{\mathcal{C}}$ is an $\infty$-category and the map

$u_{f,f'}: (Y \times X' ) \coprod _{ (X \times X')} (X \times Y' ) \hookrightarrow Y \times Y'$

is inner anodyne (Lemma 1.5.7.5). $\square$

Proof of Theorem 1.5.7.1. Let $G$ be a graph and let $\operatorname{\mathcal{C}}$ be an $\infty$-category; we wish to show that the canonical map

$\operatorname{Fun}( \operatorname{N}_{\bullet }(\operatorname{Path}[G] ), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( G_{\bullet }, \operatorname{\mathcal{C}})$

is a trivial Kan fibration. This follows from Proposition 1.5.7.6, since the inclusion $G_{\bullet } \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ is inner anodyne (Proposition 1.5.7.3). $\square$

Before giving the proof of Proposition 1.5.7.3, let us illustrate its contents with some examples.

Example 1.5.7.7 (The Spine of a Simplex). Let $n \geq 0$ and let $\Delta ^{n}$ be the standard $n$-simplex (Example 1.1.0.9). We let $\operatorname{Spine}[n]$ denote the simplicial subset of $\Delta ^{n}$ whose $k$-simplices are monotone maps $\sigma : [k] \rightarrow [n]$ satisfying $\sigma (k) \leq \sigma (0) + 1$. We will refer to $\operatorname{Spine}[n]$ as the spine of the simplex $\Delta ^{n}$. More informally, it is comprised of all vertices of $\Delta ^{n}$, together with those edges which join adjacent vertices. The spine $\operatorname{Spine}[n]$ is a simplicial set of dimension $\leq 1$, which we can identify with the directed graph $G$ depicted in the diagram

$\xymatrix@R =50pt@C=50pt{ 0 \ar [r] & 1 \ar [r] & 2 \ar [r] & \cdots \ar [r] & n. }$

Under this identification, the map $u: G_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ corresponds to the inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ (see Example 1.3.7.2). Invoking Proposition 1.5.7.3 and Theorem 1.5.7.1, we obtain the following:

$(a)$

The inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ is inner anodyne.

$(b)$

For any $\infty$-category $\operatorname{\mathcal{C}}$, the restriction map $\operatorname{Fun}( \Delta ^ n, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \operatorname{Spine}[n], \operatorname{\mathcal{C}})$ is a trivial Kan fibration.

Remark 1.5.7.8 (The Generalized Associative Law). Let $\operatorname{\mathcal{C}}$ be an ordinary category and let $n \geq 0$ be an integer. Applying Remark 1.5.7.4 to the inner anodyne inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ of Example 1.5.7.7, we deduce that every diagram

$X_0 \xrightarrow {f_1} X_1 \xrightarrow { f_2} X_2 \rightarrow \cdots \xrightarrow {f_ n} X_ n$

can be extended uniquely to a functor $[n] \rightarrow \operatorname{\mathcal{C}}$. In particular, it shows that $\operatorname{\mathcal{C}}$ satisfies the “generalized associative law”: the iterated composition $f_{n} \circ f_{n-1} \circ \cdots \circ f_{2} \circ f_{1}$ is well-defined (that is, it does not depend on a choice of parenthesization). In essence, Proposition 1.5.7.3 can be regarded as an extension of this generalized associative law to the setting of $\infty$-categories.

Corollary 1.5.7.9. Let $S$ be a simplicial set. Then $S$ is isomorphic to the nerve of a category if and only if it satisfies the following condition for every integer $n \geq 0$:

$(\ast _ n)$

Every morphism of simplicial sets $\operatorname{Spine}[n] \rightarrow S$ extends uniquely to an $n$-simplex of $S$.

Proof. Assume that $S$ satisfies condition $(\ast _ n)$ for each $n \geq 0$; we will show that $S$ is isomorphic to the nerve of a category (the reverse implication follows from Remark 1.5.7.8). By virtue of Proposition 1.3.4.1, it will suffice to show that for $0 < i < n$, every inner horn $\sigma : \Lambda ^{n}_{i} \rightarrow S$ can be extended to an $n$-simplex of $S$. Note that $\Lambda ^{n}_{i}$ contains the spine $\operatorname{Spine}[n]$. Applying condition $(\ast _ n)$, we deduce that there is a unique $n$-simplex $\sigma '$ of $S$ satisfying $\sigma '|_{ \operatorname{Spine}[n] } = \sigma |_{ \operatorname{Spine}[n] }$. We will complete the proof by showing that $\sigma = \sigma '|_{ \Lambda ^{n}_{i} }$. If $n = 2$, then $\Lambda ^{n}_{i} = \operatorname{Spine}[n]$ and there is nothing to prove. We may therefore assume that $n > 2$, so that $\Lambda ^{n}_{i}$ contains the $1$-skeleton of $\Delta ^ n$. For every pair of integers $0 \leq j \leq k \leq n$, let $e_{j,k}$ denote the corresponding edge of $\Delta ^ n$. Using condition $(\ast _{n-1})$, we are reduced to showing that $\sigma ( e_{j,k} ) = \sigma '( e_{j,k} )$ for every pair of integers $0 \leq j \leq k \leq n$. We proceed by induction on the difference $k - j$. If $k - j \leq 1$, then $e_{j,k}$ is contained in the spine $\operatorname{Spine}[n]$ and there is nothing to prove. Otherwise, we can choose an integer $\ell$ satisfying $j < \ell < k$. Let $\tau$ denote the $2$-simplex of $\Delta ^ n$ given by the triple $( j < \ell < k )$. Replacing $\ell$ by $i$ if necessary, we can arrange that $\tau$ is contained in the horn $\Lambda ^{n}_{i}$. Our inductive hypothesis guarantees that $\sigma$ and $\sigma '$ coincide on the edges $e_{j, \ell }$ and $e_{\ell ,k}$. Invoking $(\ast _2)$, we conclude that $\sigma \circ \tau = \sigma ' \circ \tau$, so that $\sigma$ and $\sigma '$ also agree on the edge $e_{j,k}$. $\square$

Remark 1.5.7.10. Let $\operatorname{\mathcal{C}}$ be an $\infty$-category and let $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ denote its homotopy category (Definition 1.4.5.3). Then the canonical map $\operatorname{\mathcal{C}}\rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ is an epimorphism of simplicial sets: that is, it induces a surjection on $n$-simplices for each $n \geq 0$. To prove this, we note that there is a commutative diagram

$\xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{n}, \operatorname{\mathcal{C}}) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{n}, \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}} )) \ar [d]^{\sim } \\ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Spine}[n], \operatorname{\mathcal{C}}) \ar [r] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Spine}[n], \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}}) ), }$

where the left vertical map is surjective (Example 1.5.7.7) and the right vertical map is bijective (Remark 1.5.7.8). It therefore suffices to show that the bottom horizontal map is surjective: that is, every sequence of composable morphisms

$X_0 \xrightarrow {f_1} X_1 \xrightarrow {f_2} X_2 \xrightarrow {f_3} X_3 \rightarrow \cdots \xrightarrow {f_ n} X_ n$

in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ can be lifted to a sequence of composable morphisms in $\operatorname{\mathcal{C}}$, which is immediate from the definition of $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.

Example 1.5.7.11 (The Simplicial Circle). Let $\Delta ^1 / \operatorname{\partial \Delta }^1$ denote the simplicial set obtained from $\Delta ^1$ by collapsing the boundary $\operatorname{\partial \Delta }^1$ to a point, so that we have a pushout diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^1 \ar [r] \ar [d] & \Delta ^1 \ar [d] \\ \Delta ^0 \ar [r] & \Delta ^1 / \operatorname{\partial \Delta }^1. }$

We will refer to $\Delta ^1 / \operatorname{\partial \Delta }^1$ as the simplicial circle; note that the geometric realization $| \Delta ^1 / \operatorname{\partial \Delta }^1 |$ is isomorphic to the standard circle $S^1$ as a topological space. The simplicial set $\Delta ^1 / \operatorname{\partial \Delta }^1$ has dimension $\leq 1$, and can therefore be identified with the directed graph $G$ depicted in the diagram

$\xymatrix@R =50pt@C=50pt{ \bullet \ar@ (ur,ul)[] }$

Note that the path category $\operatorname{Path}[G]$ can be identified with the category $B\operatorname{\mathbf{Z}}_{\geq 0}$ associated to the monoid $\operatorname{\mathbf{Z}}_{\geq 0}$ of nonnegative numbers under addition (Example 1.3.7.4) whose nerve is the simplicial set $B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}$ of Construction 1.3.2.5. Invoking Proposition 1.5.7.3 and Theorem 1.5.7.1, we obtain the following:

$(a)$

The inclusion of simplicial sets $\Delta ^1 / \operatorname{\partial \Delta }^1 \hookrightarrow B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}$ is inner anodyne.

$(b)$

For any $\infty$-category $\operatorname{\mathcal{C}}$, the restriction map $\operatorname{Fun}( B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Delta ^1 / \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}})$ is a trivial Kan fibration.

If $\operatorname{\mathcal{C}}$ is an $\infty$-category, then a morphism of simplicial sets $\Delta ^1 / \operatorname{\partial \Delta }^1 \rightarrow \operatorname{\mathcal{C}}$ can be identified with a pair $(X,f)$, where $X$ is an object of $\operatorname{\mathcal{C}}$ and $f: X \rightarrow X$ is an endomorphism of $X$ (Definition 1.4.1.5). Theorem 1.5.7.1 then guarantees that the pair $(X,f)$ can be extended to a functor of $\infty$-categories $B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0} \rightarrow \operatorname{\mathcal{C}}$.

Example 1.5.7.12 (Free Monoids). Let $M$ be the free monoid generated by a set $E$. Then we can identify $BM$ with the path category $\operatorname{Path}[G]$ of a directed graph $G$ satisfying

$\operatorname{Vert}(G) = \{ x\} \quad \quad \operatorname{Edge}(G) = E;$

see Example 1.3.7.3. Invoking Proposition 1.5.7.3 and Theorem 1.5.7.1, we obtain the following:

$(a)$

The inclusion of simplicial sets $G_{\bullet } \hookrightarrow B_{\bullet }M$ is inner anodyne.

$(b)$

For any $\infty$-category $\operatorname{\mathcal{C}}$, the restriction map $\operatorname{Fun}( B_{\bullet }M , \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( G_{\bullet }, \operatorname{\mathcal{C}})$ is a trivial Kan fibration.

Note that if $\operatorname{\mathcal{C}}$ is an $\infty$-category, then a map of simplicial sets $\sigma _0: G_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ can be identified with a choice of object $X \in \operatorname{\mathcal{C}}$ together with a collection of morphisms $\{ f_{e}: X \rightarrow X \} _{e \in E}$ indexed by $E$. It follows from $(b)$ that any such map admits an (essentially unique) extension to a functor $\sigma : B_{\bullet } M \rightarrow \operatorname{\mathcal{C}}$, which we can interpret as an action of the monoid $M$ on the object $X \in \operatorname{\mathcal{C}}$.

Proof of Proposition 1.5.7.3. Let $G$ be a directed graph and let $\operatorname{Path}[G]$ denote its path category. By definition, a morphism from $x \in \operatorname{Vert}(G)$ to $y \in \operatorname{Vert}(G)$ in the category $\operatorname{Path}[G]$ is given by a sequence of edges $\vec{e} = (e_ m, e_{m-1}, \ldots , e_1)$ satisfying

$s(e_1) = x \quad \quad t(e_{i}) = s(e_{i+1} ) \quad \quad t(e_ m) = y.$

In this case, we will refer to $m$ as the length of the morphism $\vec{e}$ and write $m = \ell (\vec{e})$. If $\sigma : \Delta ^ n \rightarrow \operatorname{N}_{\bullet }(\operatorname{Path}[G])$ is an $n$-simplex given by a diagram

$x_0 \xrightarrow { \vec{e}_1 } x_1 \xrightarrow { \vec{e}_2 } \cdots \xrightarrow { \vec{e}_ n} x_ n$

in $\operatorname{Path}[G]$, we define the length $\ell (\sigma )$ to be the sum $\ell ( \vec{e}_1 ) + \cdots + \ell ( \vec{e}_ n ) = \ell ( \vec{e}_ n \circ \cdots \circ \vec{e}_1 )$. For each positive integer $k$, let $\operatorname{N}_{\bullet }^{\leq k}( \operatorname{Path}[G] )$ denote the simplicial subset of $\operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ consisting of those simplices having length $\leq k$. We then have inclusions

$\operatorname{N}_{\bullet }^{\leq 1}( \operatorname{Path}[G] ) \subseteq \operatorname{N}_{\bullet }^{\leq 2}( \operatorname{Path}[G] ) \subseteq \operatorname{N}_{\bullet }^{\leq 3}( \operatorname{Path}[G] ) \subseteq \operatorname{N}_{\bullet }^{\leq 4}( \operatorname{Path}[G] ) \subseteq \cdots ,$

where $\operatorname{N}_{\bullet }^{\leq 1}( \operatorname{Path}[G] ) = G_{\bullet }$ and $\operatorname{N}_{\bullet }( \operatorname{Path}[G] ) = \bigcup \operatorname{N}_{\bullet }^{\leq k}( \operatorname{Path}[G] )$. Consequently, to show that the inclusion $G_{\bullet } \hookrightarrow \operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ is inner anodyne, it will suffice to show that each of the inclusion maps $\operatorname{N}_{\bullet }^{\leq k}(\operatorname{Path}[G]) \hookrightarrow \operatorname{N}_{\bullet }^{\leq k+1}(\operatorname{Path}[G] )$ is inner anodyne.

We henceforth regard the integer $k \geq 1$ as fixed. Let $\sigma : \Delta ^ n \rightarrow \operatorname{N}_{\bullet }(\operatorname{Path}[G])$ be an $n$-simplex of $\operatorname{N}_{\bullet }( \operatorname{Path}[G] )$ having length $k+1$, corresponding to a diagram

$x_0 \xrightarrow { \vec{e}_1 } x_1 \xrightarrow { \vec{e}_2 } \cdots \xrightarrow { \vec{e}_ n} x_ n$

as above. Note that $\sigma$ is nondegenerate if and only if each $\vec{e}_ i$ has positive length. We will say that $\sigma$ is normalized if it is nondegenerate and $\ell ( \vec{e}_1 ) = 1$. Let $S(n)$ be the collection of all normalized $n$-simplices of $\operatorname{N}_{\bullet }^{\leq k+1}( \operatorname{Path}[G] )$ having length $k+1$. We make the following observations:

$(i)$

If $\sigma$ belongs to $S(n)$, then the faces $d^{n}_0(\sigma )$ and $d^{n}_ n(\sigma )$ have length $\leq k$, and are therefore contained in $\operatorname{N}_{\bullet }^{\leq k}( \operatorname{Path}[G] )$.

$(ii)$

If $\sigma$ belongs to $S(n)$ and $1 < i < n$, then the face $d^{n}_ i(\sigma )$ is a normalized $(n-1)$-simplex of $\operatorname{N}_{\bullet }^{\leq k+1}( \operatorname{Path}[G] )$ of length $k+1$, and therefore belongs to $S(n-1)$.

$(iii)$

If $\sigma$ belongs to $S(n)$, then the face $d^{n}_1(\sigma )$ is not normalized. Moreover, the construction $\sigma \mapsto d^{n}_1(\sigma )$ induces a bijection from $S(n)$ to the collection of $(n-1)$-simplices of $\operatorname{N}_{\bullet }^{\leq k+1}( \operatorname{Path}[G] )$ which are nondegenerate, of length $k+1$, and not normalized.

For each $n \geq 1$, let $X(n)$ denote the simplicial subset of $\operatorname{N}_{\bullet }^{\leq k+1}( \operatorname{Path}[G] )$ given by the union of the $(n-1)$-skeleton $\operatorname{sk}_{n-1}( \operatorname{N}_{\bullet }^{\leq k+1}( \operatorname{Path}[G] ))$, the simplicial set $\operatorname{N}_{\bullet }^{\leq k}( \operatorname{Path}[G] )$, and the collection of normalized $n$-simplices of $\operatorname{N}_{\bullet }^{\leq k+1}( \operatorname{Path}[G] )$. We have inclusions

$X(1) \subseteq X(2) \subseteq X(3) \subseteq X(4) \subseteq \cdots ,$

where $\operatorname{N}_{\bullet }^{\leq k}( \operatorname{Path}[G] ) = X(1)$ and $\operatorname{N}_{\bullet }^{\leq k+1}( \operatorname{Path}[G] ) = \bigcup _{n} X(n)$. It will therefore suffice to show that the inclusion maps $X(n-1) \hookrightarrow X(n)$ are inner anodyne for $n \geq 2$. We conclude by observing that $(i)$, $(ii)$, and $(iii)$ guarantee the existence of a pushout diagram of simplicial sets

$\xymatrix@R =50pt@C=50pt{ \coprod _{\sigma \in S(n)} \Lambda ^{n}_{1} \ar [r] \ar [d] & \coprod _{\sigma \in S(n)} \Delta ^ n \ar [d] \\ X(n-1) \ar [r] & X(n). }$
$\square$