By definition, a morphism of simplicial sets $q: X \rightarrow S$ is an inner fibration if it is weakly right orthogonal to every inner horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$. From this, one can immediately deduce a stronger lifting property.
Proposition 4.1.3.1. Let $q: X \rightarrow S$ be a morphism of simplicial sets. Then $q$ is an inner fibration if and only if it satisfies the following condition:
For every square diagram of simplicial sets
where $i$ is inner anodyne, there exists a dotted arrow rendering the diagram commutative.
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X \ar [d]^{q} \\ B \ar [r] \ar@ {-->}[ur] & S } \]
Proof. The “if” direction is immediate from the definition, since the horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ is inner anodyne for $0 < i < n$. The reverse implication follows from Proposition 1.5.4.13. $\square$
Proposition 4.1.3.2. Let $f: X_{} \rightarrow Y_{}$ be a morphism of simplicial sets. Then $f$ can be factored as a composition $X_{} \xrightarrow {f'} Q_{}(f) \xrightarrow {f''} Y_{}$, where $f''$ is an inner fibration and $f'$ is inner anodyne. Moreover, the simplicial set $Q_{}(f)$ (and the morphisms $f'$ and $f''$) can be chosen to depend functorially on $f$, in such a way that the functor commutes with filtered colimits.
\[ \operatorname{Fun}( [1], \operatorname{Set_{\Delta }}) \rightarrow \operatorname{Set_{\Delta }}\quad \quad (f: X_{} \rightarrow Y_{} ) \rightarrow Q_{}(f) \]
Proof. We proceed as in the proof of Proposition 3.1.7.1. We construct a sequence of simplicial sets $\{ X(m)_{} \} _{m \geq 0}$ and morphisms $f(m): X(m)_{} \rightarrow Y_{}$ by recursion. Set $X(0)_{} = X_{}$ and $f(0) = f$. Assuming that $f(m): X(m)_{} \rightarrow Y_{}$ has been defined, let $S(m)$ denote the set of all commutative diagrams $\sigma :$
where $0 < i < n$ and the left vertical map is the inclusion. For every such commutative diagram $\sigma $, let $C_{\sigma } = \Lambda ^ n_{i}$ denote the upper left hand corner of the diagram $\sigma $, and $D_{\sigma } = \Delta ^ n$ the lower left hand corner. Form a pushout diagram
and let $f(m+1): X(m+1)_{} \rightarrow Y_{}$ be the unique map whose restriction to $X(m)_{}$ is equal to $f(m)$ and whose restriction to each $D_{\sigma }$ is equal to $u_{\sigma }$. By construction, we have a direct system of inner anodyne morphisms
Set $Q_{}(f) = \varinjlim _{m} X(m)_{}$. Then the natural map $f': X_{} \rightarrow Q_{}(f)$ is inner anodyne (since the collection of inner anodyne maps is closed under transfinite composition), and the system of morphisms $\{ f(m) \} _{m \geq 0}$ can be amalgamated to a single map $f'': Q_{}(f) \rightarrow Y_{}$ satisfying $f = f'' \circ f'$. It is clear from the definition that the construction $f \mapsto Q_{}(f)$ is functorial and commutes with filtered colimits. To complete the proof, it will suffice to show that $f''$ is a inner fibration: that is, that every lifting problem $\sigma :$
admits a solution (provided that $0 < i < n$). Let us abuse notation by identifying each $X(m)_{}$ with its image in $Q_{}(f)$. Since $\Lambda ^{n}_{i}$ is a finite simplicial set, its image under $v$ is contained in $X(m)_{}$ for some $m \gg 0$. In this case, we can identify $\sigma $ with an element of the set $S(m)$, so that the lifting problem
admits a solution by construction. $\square$
Applying Proposition 4.1.3.2 in the special case $Y = \Delta ^0$, we obtain the following:
Corollary 4.1.3.3. Let $X$ be a simplicial set. Then there exists an inner anodyne morphism $f: X \rightarrow Q(X)$, where $Q(X)$ is an $\infty $-category. Moreover, the $\infty $-category $Q(X)$ (and the morphism $f$) can be chosen to depend functorially on $X$, in such a way that the functor $X \mapsto Q(X)$ commutes with filtered colimits.
Using Proposition 4.1.3.2, we obtain the following counterpart of Proposition 4.1.3.1:
Corollary 4.1.3.4. Let $i: A \rightarrow B$ be a morphism of simplicial sets. Then $i$ is inner anodyne if and only if it satisfies the following condition:
For every square diagram of simplicial sets
where $q$ is an inner fibration, there exists a dotted arrow rendering the diagram commutative.
\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{i} \ar [r] & X \ar [d]^{q} \\ B \ar [r] \ar@ {-->}[ur] & S } \]
Proof. The “if” direction follows from Proposition 4.1.3.1. For the converse, suppose that condition $(\ast )$ is satisfied. Using Proposition 4.1.3.2, we can factor $i$ as a composition $A \xrightarrow {i'} Q \xrightarrow {q} B$, where $i'$ is inner anodyne and $q$ is an inner fibration. If the lifting problem
admits a solution, then the morphism $r$ exhibits $i$ as a retract of $i'$ (in the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$). Since the collection of inner anodyne morphisms is closed under retracts, it follows that $i$ is inner anodyne. $\square$