Theorem 2.4.5.1 (Cordier-Porter). Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category. If $\operatorname{\mathcal{C}}_{\bullet }$ is locally Kan, then the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$ is an $\infty $-category.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
2.4.5 From Simplicial Categories to $\infty $-Categories
Our goal in this section is to prove the following result (see [MR838654]):
The proof of Theorem 2.4.5.1 will require some preliminaries. We begin by analyzing the relationship of the simplicial path category $\operatorname{Path}[ \Delta ^ n ]_{\bullet } \simeq \operatorname{Path}[n]_{\bullet }$ with the subcategory $\operatorname{Path}[ \Lambda ^{n}_{i} ]_{\bullet }$, where $\Lambda ^{n}_{i} \subseteq \Delta ^ n$ is an inner horn.
Notation 2.4.5.2 (Cubes as Simplicial Sets). Let $I$ be a set. We let $\operatorname{\raise {0.1ex}{\square }}^{I}$ denote the simplicial set given by the product $\prod _{i \in I} \Delta ^1$. We will refer to $\operatorname{\raise {0.1ex}{\square }}^{I}$ as the $I$-cube. Equivalently, we can describe $\operatorname{\raise {0.1ex}{\square }}^{I}$ as the nerve of the power set $P(I) = \{ I_0 \subseteq I \} $, where we regard $P(I)$ as partially ordered with respect to inclusion. In the special case where $I$ is the set $\{ 1, 2, \ldots , n \} $ for some nonnegative integer $n$, we will denote the simplicial set $\operatorname{\raise {0.1ex}{\square }}^{I}$ by $\operatorname{\raise {0.1ex}{\square }}^{n}$ and refer to it as the standard $n$-cube.
Remark 2.4.5.3. Let $I$ be a finite set and let $\operatorname{\raise {0.1ex}{\square }}^{I}$ be the $I$-cube of Notation 2.4.5.2. Then the geometric realization $| \operatorname{\raise {0.1ex}{\square }}^{I} |$ can be identified with the topological space $\prod _{i \in I} [0,1]$. In particular, the geometric realization $| \operatorname{\raise {0.1ex}{\square }}^{n} |$ is homeomorphic to the standard cube This is a tautology in the case $n = 1$, and follows in general from the compatibility of geometric realizations with products of finite simplicial sets (see Corollary 3.6.2.2).
\[ \{ (t_1, t_2, \ldots , t_ n ) \in \operatorname{\mathbf{R}}^{n}: 0 \leq t_ i \leq 1 \} . \]
Remark 2.4.5.4. Let $n \geq 0$ be a nonnegative integer. For every pair of integers $0 \leq i < j \leq n$, we can identify morphisms from $i$ to $j$ in the path category $\operatorname{Path}[n]$ with subsets $S \subseteq [n]$ having least element $i$ and largest element $j$. The construction $S \mapsto ( \{ i, i+1, \ldots , j-1, j \} \setminus S )$ then induces a bijection $\operatorname{Hom}_{ \operatorname{Path}[n]}(i,j) \simeq P( \{ i+1, i+2, \ldots , j-2, j-1 \} )$, which extends to uniquely to an isomorphism of simplicial sets In particular, we have a canonical isomorphism of simplicial sets $\operatorname{Hom}_{ \operatorname{Path}[n] }(0,n)_{\bullet } \simeq \operatorname{\raise {0.1ex}{\square }}^{n-1}$. Under these isomorphisms, the composition law on $\operatorname{Path}[n]_{\bullet }$ is given for $i < j < k$ by the construction
\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Path}[n] }(i, j )_{\bullet } & \simeq & \operatorname{N}_{\bullet }( P( \{ i+1, i+2, \ldots , j-2, j-1\} ) ) \\ & \simeq & \operatorname{\raise {0.1ex}{\square }}^{ \{ i+1, i+2, \ldots , j-2, j-1 \} } \\ & \simeq & \operatorname{\raise {0.1ex}{\square }}^{j-i-1}. \end{eqnarray*}
\begin{eqnarray*} \operatorname{Hom}_{\operatorname{Path}[n] }(j, k )_{\bullet } \times \operatorname{Hom}_{\operatorname{Path}[n] }(i, j )_{\bullet } & \simeq & \operatorname{\raise {0.1ex}{\square }}^{ \{ j+1, \ldots , k-1 \} } \times \operatorname{\raise {0.1ex}{\square }}^{ \{ i+1, \ldots , j-1 \} } \\ & \simeq & \operatorname{\raise {0.1ex}{\square }}^{ \{ j+1, \ldots , k-1 \} } \times \{ 0\} \times \operatorname{\raise {0.1ex}{\square }}^{ \{ i+1, \ldots , j-1 \} } \\ & \hookrightarrow & \operatorname{\raise {0.1ex}{\square }}^{ \{ j+1, \ldots , k-1 \} } \times \Delta ^1 \times \operatorname{\raise {0.1ex}{\square }}^{ \{ i+1, \ldots , j-1 \} } \\ & \simeq & \operatorname{\raise {0.1ex}{\square }}^{ \{ i+1, \ldots , k-1 \} } \\ & \simeq & \operatorname{Hom}_{\operatorname{Path}[n] }(i, k)_{\bullet }. \end{eqnarray*}
Notation 2.4.5.5 (Subsets of the $I$-Cube). Let $I$ be a finite set and let $\operatorname{\raise {0.1ex}{\square }}^{I}$ denote the $I$-cube of Notation 2.4.5.2. For each element $i \in I$, we can identify $\operatorname{\raise {0.1ex}{\square }}^{I}$ with the product $\Delta ^1 \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} }$. Using this identification, we obtain simplicial subsets which we will refer to as faces of $\operatorname{\raise {0.1ex}{\square }}^{I}$. The (disjoint) union of these two faces is another simplicial subset of $\operatorname{\raise {0.1ex}{\square }}^{I}$, which we can identify with the product $\partial \Delta ^1 \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} }$. We let $\operatorname{\partial \raise {0.1ex}{\square }}^{I}$ denote the simplicial subset of $\operatorname{\raise {0.1ex}{\square }}^{I}$ given by the union of all its faces. We will refer to $\operatorname{\partial \raise {0.1ex}{\square }}^{I}$ as the boundary of the $I$-cube $\operatorname{\raise {0.1ex}{\square }}^{I}$. For $i \in I$, we let $\boldsymbol {\sqcap }^{I}_{i} \subseteq \operatorname{\raise {0.1ex}{\square }}^{I}$ denote the simplicial subset of $\operatorname{\raise {0.1ex}{\square }}^{I}$ given by the union of the face $(\{ 0\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } )$ with $\bigcup _{j \in I \setminus \{ i\} } ( \partial \Delta ^1 \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ j\} })$. Similarly, we let $\boldsymbol {\sqcup }^{I}_{i}$ denote the simplicial subset of $\operatorname{\raise {0.1ex}{\square }}^{I}$ given by the union of the face $(\{ 1\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } )$ with $\bigcup _{j \in I \setminus \{ i\} } ( \partial \Delta ^1 \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ j\} })$. We will refer to the simplicial subsets $\boldsymbol {\sqcap }^{I}_{i}, \boldsymbol {\sqcup }^{I}_{i} \subseteq \operatorname{\raise {0.1ex}{\square }}^{I}$ as hollow $I$-cubes. In the special case where $I = \{ 1, \ldots , n \} $ for some nonnegative integer $n$, we will denote the simplicial sets $\operatorname{\partial \raise {0.1ex}{\square }}^{I}$, $\boldsymbol {\sqcap }^{I}_{i}$, and $\boldsymbol {\sqcup }^{I}_{i}$ by $\operatorname{\partial \raise {0.1ex}{\square }}^{n}$, $\boldsymbol {\sqcap }^{n}_{i}$, and $\boldsymbol {\sqcup }^{n}_{i}$, respectively.
\[ \{ 0\} \times \operatorname{\raise {0.1ex}{\square }}^{ I \setminus \{ i\} } \subseteq \operatorname{\raise {0.1ex}{\square }}^{I} \supseteq \{ 1\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } \]
\[ \bigcup _{i \in I} ( \partial \Delta ^1 \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ) \]
Remark 2.4.5.6. Roughly speaking, one can think of the simplicial set $\operatorname{\partial \raise {0.1ex}{\square }}^{n}$ as obtained from the $n$-cube $\operatorname{\raise {0.1ex}{\square }}^{n}$ by removing its interior, while the subsets $\boldsymbol {\sqcap }^{n}_{i}, \boldsymbol {\sqcup }^{n}_{i}$ are obtained from $\operatorname{\raise {0.1ex}{\square }}^{n}$ by removing the interior together with a single face.
Example 2.4.5.7. The standard $2$-cube $\operatorname{\raise {0.1ex}{\square }}^2 \simeq \Delta ^1 \times \Delta ^1$ is depicted in the diagram It is a simplicial set of dimension $2$, having exactly two nondegenerate $2$-simplices (represented by the triangular regions in the preceding diagram) and five nondegenerate edges. The boundary $\operatorname{\partial \raise {0.1ex}{\square }}^2$ is a $1$-dimensional simplicial subset of $\operatorname{\raise {0.1ex}{\square }}^{2}$, obtained by removing the nondegenerate $2$-simplices along with the “internal” edge to obtain the directed graph depicted in the diagram Each of the hollow $2$-cubes $\boldsymbol {\sqcap }^{2}_{1}, \boldsymbol {\sqcap }^{2}_{2}, \boldsymbol {\sqcup }^{2}_{1}, \boldsymbol {\sqcup }^{2}_{2}$ can be obtained from $\operatorname{\partial \raise {0.1ex}{\square }}^{2}$ by further deletion of a single edge, represented in the diagrams
\[ \xymatrix@R =50pt@C=50pt{ \bullet \ar [r] \ar [d] \ar [dr] & \bullet \ar [d] \\ \bullet \ar [r] & \bullet . } \]
\[ \xymatrix@R =50pt@C=50pt{ \bullet \ar [r] \ar [d] & \bullet \ar [d] \\ \bullet \ar [r] & \bullet . } \]
\[ \xymatrix@R =50pt@C=50pt{ \bullet \ar [r] \ar [d] \ar@ {}[dr]|{\boldsymbol {\sqcap }^{2}_{1}} & \bullet & \bullet \ar [r] \ar [d] \ar@ {}[dr]|{\boldsymbol {\sqcap }^{2}_{2}} & \bullet \ar [d] \\ \bullet \ar [r] & \bullet & \bullet & \bullet \\ \bullet \ar [r] \ar@ {}[dr]|{\boldsymbol {\sqcup }^{2}_{1}} & \bullet \ar [d] & \bullet \ar [d] \ar@ {}[dr]|{\boldsymbol {\sqcup }^{2}_{2}} & \bullet \ar [d] \\ \bullet \ar [r] & \bullet & \bullet \ar [r] & \bullet . } \]
Proposition 2.4.5.8. Let $0 < i < n$ be positive integers and let $F: \operatorname{Path}[ \Lambda ^{n}_{i} ]_{\bullet } \rightarrow \operatorname{Path}[ \Delta ^ n ]_{\bullet }$ be the simplicial functor induced by the horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$. Then:
The functor $F$ is bijective on objects; in particular, we can identify the objects of $\operatorname{Path}[ \Lambda ^{n}_{i} ]_{\bullet }$ with elements of the set $[n] = \{ 0 < 1 < \cdots < n \} $.
For $(j,k) \neq (0,n)$, the functor $F$ induces an isomorphism of simplicial sets
The functor $F$ induces a monomorphism of simplicial sets
whose image can be identified with the hollow cube
introduced in Notation 2.4.5.5.
\[ \operatorname{Hom}_{ \operatorname{Path}[ \Lambda ^{n}_{i} ] }(j, k)_{\bullet } \simeq \operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n ] }(j, k)_{\bullet }. \]
\[ \operatorname{Hom}_{ \operatorname{Path}[ \Lambda ^{n}_{i} ] }(0,n)_{\bullet } \hookrightarrow \operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n ] }(0,n)_{\bullet }, \]
\[ {\boldsymbol {\sqcap }}^{n-1}_{i} \subseteq \operatorname{\raise {0.1ex}{\square }}^{n-1} \simeq \operatorname{Hom}_{ \operatorname{Path}[\Delta ^ n]}(0,n)_{\bullet } \]
Proof. Assertion $(a)$ is immediate from Theorem 2.4.4.10. To prove $(b)$ and $(c)$, fix an integer $m \geq 0$. Using Lemma 2.4.4.16, we see that $\operatorname{Path}[ \Delta ^ n ]_{m}$ can be identified with the path category $\operatorname{Path}[G]$ of a directed graph $G$ which can be described concretely as follows:
The vertices of $G$ are the elements of the set $[n] = \{ 0 < 1 < \cdots < n \} $.
For $0 \leq j < k \leq n$, an edge of $G$ with source $j$ and target $k$ is a chain of subsets
\[ \{ j, j+1, \ldots , k-1, k\} \supseteq I_0 \supseteq \cdots \supseteq I_ m = \{ j, k \} \]
Using Theorem 2.4.4.10, we see that $\operatorname{Path}[ \Lambda ^ n_ i ]_{m}$ can be identified with the path category of the directed subgraph $G' \subseteq G$ having the same vertices, where an edge $\overrightarrow {I} = (I_0 \supseteq \cdots \supseteq I_ m)$ of $G$ belongs to $G'$ if and only if the subset $I_0 \subseteq [n]$ corresponds to a simplex of $\Delta ^ n$ which belongs to the horn $\Lambda ^{n}_{i}$: that is, if and only if $[n] \setminus \{ i\} \nsubseteq I_{0}$. In particular, we see that for $(j,k) \neq (0,n)$, every edge of $G$ with source $j$ and target $k$ is contained in $G'$. It follows that the simplicial functor $F$ induces a bijection $\operatorname{Hom}_{ \operatorname{Path}[\Lambda ^{n}_{i} ] }( j, k )_{m} \rightarrow \operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n] }(j, k)_{m}$ for $(j, k) \neq (0,n)$, which proves $(b)$. Moreover, the map $\operatorname{Hom}_{ \operatorname{Path}[\Lambda ^{n}_{i} ] }( 0, n )_{m} \rightarrow \operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n] }(0,n)_{m}$ is a monomorphism, whose image consists of those chains
\[ \overrightarrow {I} = (I_0 \supseteq I_1 \supseteq \cdots \supseteq I_{m}) \]
where either $I_ m \neq \{ 0,n\} $ or $([n] \setminus \{ i\} ) \nsubseteq I_0$. Under the identification of $\operatorname{Hom}_{ \operatorname{Path}[ \Delta ^ n] }(0,n)_{\bullet }$ with the cube $\operatorname{\raise {0.1ex}{\square }}^{n-1} \simeq \operatorname{N}_{\bullet }( P( \{ 1, \ldots , n-1\} ))$ described in Remark 2.4.5.4, this corresponds to collection of $m$-simplices of $\operatorname{\raise {0.1ex}{\square }}^{n-1}$ given by chains of subsets
\[ J_0 \subseteq J_1 \subseteq \cdots \subseteq J_{m} \subseteq \{ 1, \ldots , n-1 \} \]
where either $J_0 \nsubseteq \{ i\} $ or $J_ m \subsetneq \{ 1, \ldots , n-1 \} $, which is exactly the set of $m$-simplices which belong to the hollow cube $\boldsymbol {\sqcap }^{n-1}_{i}$. $\square$
To apply Proposition 2.4.5.8, we record the following elementary observation about simplicial categories:
Proposition 2.4.5.9. Let $\operatorname{\mathcal{E}}_{\bullet }$ be a simplicial category containing a pair of objects $x,y \in \operatorname{Ob}( \operatorname{\mathcal{E}}_{\bullet } )$. Assume that, for each object $z \in \operatorname{Ob}( \operatorname{\mathcal{E}}_{\bullet } )$, we have Let $\operatorname{\mathcal{D}}_{\bullet } \subseteq \operatorname{\mathcal{E}}_{\bullet }$ denote a simplicial subcategory having the same objects, which satisfies unless $(a,b) = (x,y)$. Let $F: \operatorname{\mathcal{D}}_{\bullet } \rightarrow \operatorname{\mathcal{C}}_{\bullet }$ be a functor of simplicial categories carrying $x$ to an object $X = F(x)$ and $y$ to an object $Y \in F(y)$, so that $F$ induces a map of simplicial sets $F_{x,y}: \operatorname{Hom}_{\operatorname{\mathcal{D}}}(x,y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$. Then the construction $\overline{F} \mapsto \overline{F}_{x,y}$ induces a bijection
\[ \operatorname{Hom}_{\operatorname{\mathcal{E}}}( z, x)_{\bullet } = \begin{cases} \{ \operatorname{id}_ x \} & \text{ if } z = x \\ \emptyset & \text{ otherwise. } \end{cases} \quad \quad \operatorname{Hom}_{\operatorname{\mathcal{E}}}( y, z)_{\bullet } = \begin{cases} \{ \operatorname{id}_ y \} & \text{ if } z = y \\ \emptyset & \text{ otherwise. } \end{cases} \]
\[ \operatorname{Hom}_{\operatorname{\mathcal{D}}}( a, b )_{\bullet } = \operatorname{Hom}_{\operatorname{\mathcal{E}}}( a, b )_{\bullet } \]
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Simplicial functors $\overline{F}: \operatorname{\mathcal{E}}_{\bullet } \rightarrow \operatorname{\mathcal{C}}_{\bullet }$ extending $F$} \} \ar [d]^{\sim } \\ \{ \textnormal{Maps $\lambda : \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ extending $F_{x,y}$} \} . } \]
Proof. Fix a map of simplicial sets $\lambda : \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,y)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ which extends $F_{x,y}$. We wish to show that there is a unique simplicial functor $\overline{F}: \operatorname{\mathcal{E}}_{\bullet } \rightarrow \operatorname{\mathcal{C}}_{\bullet }$ such that $F = \overline{F}|_{\operatorname{\mathcal{D}}_{\bullet }}$ and $\overline{F}_{x,y} = \lambda $. The uniqueness is clear: the simplicial functor $\overline{F}$ must coincide with $F$ on objects and satisfy $\overline{F}_{x',y'} = F_{x',y'}$ for $(x',y') \neq (x,y)$. To prove existence, one must show that this prescription defines a simplicial functor: that is, that for every triple of objects $a,b,c \in \operatorname{Ob}(\operatorname{\mathcal{E}}_{\bullet })$, the resulting diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(b,c)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{E}}}(a,b)_{\bullet } \ar [r] \ar [d]^{ \overline{F}_{a,b} \otimes \overline{F}_{b,c} } & \operatorname{Hom}_{\operatorname{\mathcal{E}}}(a,c)_{\bullet } \ar [d]^{ \overline{F}_{a,c} } \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(b), F(c) )_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(a), F(b) )_{\bullet } \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(a), F(c) )_{\bullet } } \]
is commutative. We consider several cases:
Suppose that $(a,b) = (x,y)$. If $c \neq y$, then the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(b,c)_{\bullet }$ is empty and the commutativity of the diagram is automatic. If $c = y$, then both compositions can be identified with the map
\[ \{ \operatorname{id}_ y \} \times \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,y)_{\bullet } \simeq \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,y)_{\bullet } \xrightarrow {\lambda } \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }. \]Suppose that $(b,c) = (x,y)$. If $a \neq x$, then the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(a,b)_{\bullet }$ is empty and the commutativity of the diagram is automatic. If $a = x$, then both compositions can be identified with the map
\[ \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,y)_{\bullet } \times \{ \operatorname{id}_ x \} \simeq \operatorname{Hom}_{\operatorname{\mathcal{E}}}(x,y)_{\bullet } \xrightarrow {\lambda } \operatorname{Hom}_{\operatorname{\mathcal{C}}}( X, Y)_{\bullet }. \]If neither $(a,b) = (x,y)$ or $(b,c) = (x,y)$, then the desired result follows from the commutativity of the diagram
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Hom}_{\operatorname{\mathcal{D}}}(b,c)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{D}}}(a,b)_{\bullet } \ar [r] \ar [d]^{ F_{a,b} \otimes F_{b,c} } & \operatorname{Hom}_{\operatorname{\mathcal{D}}}(a,c)_{\bullet } \ar [d]^{ F_{a,c} } \\ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(b), F(c) )_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(a), F(b) )_{\bullet } \ar [r] & \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(a), F(c) )_{\bullet } } \](since $F$ is assumed to be a simplicial functor).
It follows from Proposition 2.4.5.8 that for $0 < i < n$, the hypotheses of Proposition 2.4.5.9 are satisfied by the inclusion $\operatorname{\mathcal{D}}_{\bullet } = \operatorname{Path}[ \Lambda ^{n}_{i} ]_{\bullet } \hookrightarrow \operatorname{Path}[ \Delta ^ n ]_{\bullet } = \operatorname{\mathcal{E}}_{\bullet }$ and the objects $x = 0$ and $y = n$. We therefore obtain the following:
Corollary 2.4.5.10. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a simplicial category, let $0 < i < n$, and let $\sigma _0: \Lambda ^{n}_{i} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ be a map of simplicial sets, which we can identify with a simplicial functor $F: \operatorname{Path}[ \Lambda ^{n}_{i} ]_{\bullet } \rightarrow \operatorname{\mathcal{C}}_{\bullet }$ inducing a map of simplicial sets Then we have a canonical bijection
\[ \lambda _0: \boldsymbol {\sqcap }^{n-1}_{i} \simeq \operatorname{Hom}_{ \operatorname{Path}[ \Lambda ^ n_ i] }( 0, n)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(0), F(n) )_{\bullet }. \]
\[ \xymatrix@R =50pt@C=50pt{ \{ \textnormal{Maps $\sigma : \Delta ^ n \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$ with $\sigma _0 = \sigma |_{ \Lambda ^ n_{i}}$} \} \ar [d] \\ \{ \textnormal{Maps $\lambda : \operatorname{\raise {0.1ex}{\square }}^{n-1} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(0), F(n) )_{\bullet }$ with $\lambda _0 = \lambda |_{ \boldsymbol {\sqcap }^{n-1}_{i}}$} \} .} \]
To deduce Theorem 2.4.5.1 from Corollary 2.4.5.10, we will need the following standard characterization of Kan complexes (for a proof, see Proposition 4.4.2.1):
Theorem 2.4.5.11 (Homotopy Extension Lifting Property). Let $X_{\bullet }$ be a simplicial set. The following conditions are equivalent:
The simplicial set $X_{\bullet }$ is a Kan complex.
The inclusion of simplicial sets $\{ 0\} \hookrightarrow \Delta ^1$ induces a trivial Kan fibration $\operatorname{Fun}( \Delta ^1, X_{\bullet } ) \rightarrow \operatorname{Fun}( \{ 0\} , X_{\bullet } ) \simeq X_{\bullet }$.
The inclusion of simplicial sets $\{ 1\} \hookrightarrow \Delta ^1$ induces a trivial Kan fibration $\operatorname{Fun}( \Delta ^1, X_{\bullet } ) \rightarrow \operatorname{Fun}( \{ 1\} , X_{\bullet } ) \simeq X_{\bullet }$.
Corollary 2.4.5.12. Let $X_{\bullet }$ be a Kan complex and let $I$ be a finite set containing a distinguished element $i$. Then:
Every map of simplicial sets $f: \boldsymbol {\sqcup }^{I}_{i} \rightarrow X_{\bullet }$ can be extended to a map $\overline{f}: \operatorname{\raise {0.1ex}{\square }}^{I} \rightarrow X_{\bullet }$.
Every map of simplicial sets $g: \boldsymbol {\sqcap }^{I}_{i} \rightarrow X_{\bullet }$ can be extended to a map $\overline{g}: \operatorname{\raise {0.1ex}{\square }}^{I} \rightarrow X_{\bullet }$.
Proof. Unwinding the definitions, we see that $\boldsymbol {\sqcup }^{I}_{i}$ can be identified with the pushout
\[ (\{ 1\} \times \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } ) \coprod _{ \{ 1\} \times \operatorname{\partial \raise {0.1ex}{\square }}^{I \setminus \{ i\} } } (\Delta ^1 \times \operatorname{\partial \raise {0.1ex}{\square }}^{I \setminus \{ i\} } ). \]
Consequently, a map of simplicial sets $f: \boldsymbol {\sqcup }^{I}_{i} \rightarrow X_{\bullet }$ can be identified with a commutative diagram of solid arrows
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \raise {0.1ex}{\square }}^{I \setminus \{ i\} } \ar [r] \ar [d] & \operatorname{Fun}( \Delta ^1, X_{\bullet } ) \ar [d] \\ \operatorname{\raise {0.1ex}{\square }}^{I \setminus \{ i\} } \ar [r] \ar@ {-->}[ur] & \operatorname{Fun}( \{ 1\} , X_{\bullet } ), } \]
and an extension $\overline{f}: \operatorname{\raise {0.1ex}{\square }}^{I} \rightarrow X_{\bullet }$ of $f$ can be identified with a solution to the associated lifting problem. If $X_{\bullet }$ is a Kan complex, then the right vertical arrow is a trivial Kan fibration (Theorem 2.4.5.11), so the desired extension exists by virtue of Proposition 1.5.5.4. This proves $(a)$; the proof of $(b)$ is similar. $\square$
Proof of Theorem 2.4.5.1. Let $\operatorname{\mathcal{C}}_{\bullet }$ be a locally Kan simplicial category; we wish to show that the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$ is an $\infty $-category. Fix positive integers $0 < i < n$; we wish to show that every map of simplicial sets $\sigma _0: \Lambda ^ n_ i \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$ can be extended to an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{C}})$. Let us identify $\sigma _0$ with a simplicial functor $F: \operatorname{Path}[ \Lambda ^{n}_{i} ]_{\bullet } \rightarrow \operatorname{\mathcal{C}}_{\bullet }$ inducing a map of simplicial sets $\lambda _0: {\boldsymbol {\sqcap }}^{n-1}_{i} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(0), F(n) )_{\bullet }$. By virtue of Corollary 2.4.5.10, it will suffice to show that $\lambda _0$ can be extended to a map of simplicial sets $\lambda : \operatorname{\raise {0.1ex}{\square }}^{n-1} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(0), F(n) )_{\bullet }$. The existence of this extension follows from Corollary 2.4.5.12, by virtue of our assumption that $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( F(0), F(n) )_{\bullet }$ is a Kan complex. $\square$