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 $\operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{Path}[G] ), \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( G_{\bullet }, \operatorname{\mathcal{C}})$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
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:
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 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 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.
\[ \xymatrix@R =50pt@C=50pt{ \bullet \ar [r] & \bullet \ar [r] & \bullet . } \]
\[ \operatorname{Fun}( \Delta ^2, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \Lambda ^{2}_{1}, \operatorname{\mathcal{C}}) \]
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 is bijective. Combining this observation with Proposition 1.3.3.1, we obtain a bijection 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).
\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }( \operatorname{Path}[G] ), \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( G_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \]
\[ \operatorname{Hom}_{\operatorname{Cat}}( \operatorname{Path}[G], \operatorname{\mathcal{C}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( G_{\bullet }, \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ). \]
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 is inner anodyne.
\[ u_{f,f'}: (Y \times X' ) \coprod _{ (X \times X')} (X \times Y' ) \hookrightarrow Y \times Y' \]
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 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:
The inclusion $\operatorname{Spine}[n] \hookrightarrow \Delta ^ n$ is inner anodyne.
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.
\[ \xymatrix@R =50pt@C=50pt{ 0 \ar [r] & 1 \ar [r] & 2 \ar [r] & \cdots \ar [r] & n. } \]
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 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.
\[ X_0 \xrightarrow {f_1} X_1 \xrightarrow { f_2} X_2 \rightarrow \cdots \xrightarrow {f_ n} X_ n \]
Remark 1.5.7.8 admits a converse:
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 $n \geq 0$:
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 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 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 $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{n}, \operatorname{\mathcal{C}}) \ar [r] \ar [d] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^{n}, \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}} )) \ar [d]^{\sim } \\ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Spine}[n], \operatorname{\mathcal{C}}) \ar [r] & \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Spine}[n], \operatorname{N}_{\bullet }(\mathrm{h} \mathit{\operatorname{\mathcal{C}}}) ), } \]
\[ X_0 \xrightarrow {f_1} X_1 \xrightarrow {f_2} X_2 \xrightarrow {f_3} X_3 \rightarrow \cdots \xrightarrow {f_ n} X_ n \]
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 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 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:
The inclusion of simplicial sets $\Delta ^1 / \operatorname{\partial \Delta }^1 \hookrightarrow B_{\bullet } \operatorname{\mathbf{Z}}_{\geq 0}$ is inner anodyne.
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.
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^1 \ar [r] \ar [d] & \Delta ^1 \ar [d] \\ \Delta ^0 \ar [r] & \Delta ^1 / \operatorname{\partial \Delta }^1. } \]
\[ \xymatrix@R =50pt@C=50pt{ \bullet \ar@ (ur,ul)[] } \]
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 see Example 1.3.7.3. Invoking Proposition 1.5.7.3 and Theorem 1.5.7.1, we obtain the following:
The inclusion of simplicial sets $G_{\bullet } \hookrightarrow B_{\bullet }M$ is inner anodyne.
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.
\[ \operatorname{Vert}(G) = \{ x\} \quad \quad \operatorname{Edge}(G) = E; \]
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
\[ \xymatrix@R =50pt@C=50pt{ \coprod _{\sigma \in S(n)} \Lambda ^{n}_{1} \ar [r] \ar [d] & \coprod _{\sigma \in S(n)} \Delta ^ n \ar [d] \\ X(n-1) \ar [r] & X(n). } \]
$\square$